Operators which have a closed quasi-nilpotent part

Abstract

Abstract. We find several conditions for the quasi-nilpotent part of a bounded operator acting on a Banach space to be closed. Most of these conditions are established for semi-Fredholm operators or, more generally, for operators which admit a generalized Kato decomposition. For these operators the property of having a closed quasi-nilpotent part is related to the so-called single valued extension property. The quasi-nilpotent part of an operator and the SVEPThe single valued extension property was first introduced by Dunford An operator T ∈ L(X) is said to have the SVEP if T has the SVEP at every point λ ∈ C.

On rank one perturbations of complex symmetric operators

“…Then H 0 (T) is a linear (not necessarily closed) subspace of H. We remark from [2] that if T has the single-valued extension property, then…”

“…for all λ ∈ C. It is well known from [1] and [2] that if H 0 (T−λ) = {0} for all λ ∈ C, then T has the single-valued extension property. The analytical core K(T) of T is the set of all x ∈ H with the property that there is a sequence {u n } ⊂ H and a constant δ > 0 such that x = u 0 , Tu n+1 = u n , and u n ≤ δ n x for every integer n ≥ 0 (see [1] for more details).…”

Local Spectral Theory for R and S Satisfying RnSRn = Rj

“…Then, R j−n has SVEP, hence by Lemma 2 SR n has SVEP. Therefore, by (13) and by assumption, H 0 (λI − R j−n ) = X R j−n ({λ}) is closed for every λ ∈ C. By (13) and 3, H 0 (SR n ) = X SR n ({0}) is closed. Following the procedure of [1], let 0 = λ ∈ C; by ( [7], Proposition 3.3.1, part (f)) we have…”

“…Every multiplier of a semi-simple commutative Banach algebra has property (Q), see ( [13], Theorem 1.8), in particular every convolution operator T µ , µ ∈ M(G), on the group algebra L 1 (G) has property (Q), but there are convolution operators which do not enjoy property (C) (see [7], Chapter 4).…”

Local Spectral Properties Under Conjugations

“…Recall that an operator T ∈ L(H) is said to be algebraic if there exists a polynomial h such that h(T ) = 0. The SVEP is preserved under commuting algebraic perturbations, see [5], and is also preserved under the functional calculus, see [2,Chapter 2], i.e., if T has SVEP then f (T ) has SVEP for every f ∈ H(σ(T )). Moreover, if T is polaroid then f (T ) is polaroid for every f ∈ H(σ(T )), see [1].…”